Appendix

We assume that the Cauchy hypersurface \(\textit{S}_0\) is asymptotically Euclidean and \(\textit{A}\) is an uniformly elliptic linear differential operator with smooth coefficients such that the coefficients are bounded in any \(C^m({\mathcal S_0})\qquad \forall \, m\in {\mathbb N}\) . Then we can prove that the solutions \(f\in \mathscr {S}' (\textit{S}_0)\) of the eigenvalue problem \( Af = \mu f\) belong to \(C^\infty ({\mathcal S_0})\) . The result is well known but we give a proof for the convenience of the reader.